Olympiad problems

1998 Chile National Olympiad, 3

Evaluate $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}$.

2021 Auckland Mathematical Olympiad, 3

Tags: algebra , Nested Radicals , radical

For how many integers $n$ between $ 1$ and $2021$ does the infinite nested expression $$\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}}$$ give a rational number?

2003 VJIMC, Problem 3

Tags: limits , real analysis , Nested Radicals

Find the limit $$\lim_{n\to\infty}\sqrt{1+2\sqrt{1+3\sqrt{\ldots+(n-1)\sqrt{1+n}}}}.$$

2021 Alibaba Global Math Competition, 19

Tags: algebra , polynomial , sum of roots , roots of unity , Nested Radicals , college contests

Find all real numbers of the form $\sqrt[p]{2021+\sqrt[q]{a}}$ that can be expressed as a linear combination of roots of unity with rational coefficients, where $p$ and $q$ are (possible the same) prime numbers, and $a>1$ is an integer, which is not a $q$-th power.

2010 Laurențiu Panaitopol, Tulcea, 1

Tags: Sequences , limits , real analysis , Weierstrass , Nested Radicals

Show that if $ \left( s_n \right)_{n\ge 0} $ is a sequence that tends to $ 6, $ then, the sequence $$ \left( \sqrt[3]{s_n+\sqrt[3]{s_{n-1}+\sqrt[3]{s_{n-2}+\sqrt[3]{\cdots +\sqrt[3]{s_0}}}}} \right)_{n\ge 0} $$ tends to $ 2. $ [i]Mihai Bălună[/i]